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Astron. Astrophys. 337, 945-954 (1998)

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4. The retrieval of the nucleus

A complete model image of the comet is obtained by superimposing the nucleus on the coma model described above. The nucleus is represented by the scaled point spread function, [FORMULA]. In order to make a comparison with the observed images, the model image must be integrated over the [FORMULA] sub-pixels to recover the original pixel of the WFPC2. In performing this integration, we allowed for 64 different possibilities, corresponding to the 64 possible locations of the nucleus [FORMULA] on the sub-sampled grid. The location of the nucleus and the scaling factor [FORMULA] were determined by minimizing the residuals [FORMULA] between the model and observed images:

[EQUATION]

Note that the magnitude of the coma is not a free parameter, as that was fixed by the fitting procedure used to derive the coma model. The location of the nucleus is apparently located very accurately, as illustrated in Fig. 4, which displays the x profiles for the optimum value of [FORMULA] and the y profiles for the optimum value of [FORMULA]. A non-optimal value of [FORMULA] was used to increase the clarity in the plot but the best solution is obviously that which closely parallels the observed profiles .

[FIGURE] Fig. 4. Influence of the sub-pixel location of the nucleus: comparison of a family of solutions with the observation (thick solid line).

The [FORMULA] noise of a signal B is given by

[EQUATION]

where [FORMULA]7 electrons/DN is the gain, [FORMULA]5 electrons is the readout noise, and [FORMULA]0.01 expresses the flat field noise as a fraction of the signal. For a typical signal of 2000 DN, the [FORMULA] value of the noise is 26 DN, which is insignificant compared to the differences between the models and the data plotted in Fig. 4. Although our procedure provides excellent discrimination among different sub-pixel locations of the nucleus, we note that our determination of the nuclear magnitude is essentially unaffected by the exact location. The primary benefit of including sub-pixel sampling is to improve the fit between the model and the data (i.e., sub-sampling allows a better approximation to the skewness of the spatial brightness distribution).

An example of the results from our minimization procedure is displayed in Fig. 5, where the residual Q is plotted as a function of the three parameters [FORMULA] and [FORMULA]. [FORMULA], shown for the optimum values of the coordinates, is very well behaved and allows a precise determination of [FORMULA].

[FIGURE] Fig. 5. Variations of the residual Q with [FORMULA] (top), [FORMULA] (middle), and [FORMULA] (bottom).

Fig. 6 illustrates the details of our solution for the third observation. The different skewness of the x-profiles of the coma and the nucleus is readily explained by the fact that the nucleus is not located at the opto-center of the coma, as already discussed. As expected, the detailed variations of the coma signal illustrated in Fig. 3 are washed out by the convolution with the PSF and by the integration over the original pixel. In practice, the coma model is rather insensitive to sub-pixel brightness variations, as checked by several tests.

[FIGURE] Fig. 6. Comparison of the third observation (thick solid line) with the coma model (dash line), the nucleus model (dash-triple-dot line) and the coma + nucleus model (dash-dot line).

Fig. 7 compares the model comet with the observed image on an expanded vertical scale. Although the quality of the fit looks excellent, one must remember that the plot has a logarithmic scale. In reality, the deviations between the model and the data in the core of the image are significantly larger than what would be expected based on photon-dominated noise statistics. We suspect that the large deviations are caused by our inability to represent accurately the highly anisotropic coma of 19P/Borrelly, as our residuals are significantly smaller for comets having a more circularly symmetric surface brightness distribution. In any case, the residuals from our fits are still less than 10% of the observed peak intensities, indicating that our retrieved size for the nucleus is not significantly affected by the imperfect fitting.

[FIGURE] Fig. 7. Comparison of the third observation (thick solid line) with the coma + nucleus model (dash line). For signals larger than [FORMULA]300 DN, the 1-sigma error bar is within the width of the thick solid line. See the text for further discussion.

Absolute magnitudes for the nucleus were determined from the scaled PSFs that best matched the observations (i.e., the [FORMULA] distributions, which represent the nucleus as it would be observed by the HST in the absence of coma). Following Holtzman et al. (1995), we calculated instrumental magnitudes by integrating the scaled PSFs over an aperture having a radius of [FORMULA], so that no aperture correction is required. We did not apply a correction for the charge transfer efficiency, since the coma ensured that there was a large background signal. Nor did we make any correction for observing geometry, as the nucleus was always located near the center of the CCD. The conversion to standard (Landolt) R magnitudes, and the additional transfer to V magnitudes, required knowledge of the V-R colour of the nucleus. As we did for comet 4P/Faye, we used [FORMULA], which corresponds to an "average" reddening of 17 % per 103Å (Jewitt, 1992). The so-called reduced magnitudes V(1,1,0) and R(1,1,0), which refer to the standard conditions [FORMULA] AU and [FORMULA], were calculated assuming a linear variation of the phase function of the nucleus with the phase angle (using a coefficient of 0.04 mag/deg). These results are given in Table 1. In order to compare them with the photographic magnitudes of the "nucleus condensation" (Roemer, 1965), we first applied a standard correction of -0.8 mag to convert from photographic to V magnitudes and the above phase law to obtain reduced magnitudes. These corrected values are scattered between 15 and 13, which are well outside our range. The faintest value of 15 was obtained when the comet was at 3.5 AU and is approximately 0.7 mag brighter than our brightest value, suggesting the plausible persistence of a coma even at this large heliocentric distance. Our range of apparent nuclear magnitudes ([FORMULA] mag) is comparable to that of comet 1P/Halley, but substantially larger than that of other comets (Jewitt, 1992).

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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